[Paper Review] Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric

Jason Miller, Scott Sheffield⋆|arXiv (Cornell University)|Jul 2, 2015

Noncommutative and Quantum Gravity TheoriesPhysics and Astronomy33 references111 citations

TL;DR

This paper establishes a metric structure on the $√{8/3}$-Liouville quantum gravity (LQG) sphere using quantum Loewner evolution (QLE), proving that the time it takes QLE to grow from one point to another defines a symmetric, a.s. continuous metric. The key result is that this QLE-induced metric satisfies the triangle inequality and agrees with the Brownian map (TBM) metric, laying the foundation for proving equivalence between $√{8/3}$-LQG and TBM in subsequent works.

ABSTRACT

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $γ$, and it has long been believed that when $γ= \sqrt{8/3}$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other's structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other's structure and showing that the resulting laws agree. The present work uses a form of the quantum Loewner evolution (QLE) to construct a metric on a dense subset of a $\sqrt{8/3}$-LQG sphere and to establish certain facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric extends uniquely and continuously to the entire $\sqrt{8/3}$-LQG surface and that the resulting measure-endowed metric space is TBM.

Motivation & Objective

• To define a metric on the $√{8/3}$-LQG sphere using quantum Loewner evolution (QLE), which is expected to correspond to the Brownian map (TBM) metric.
• To show that the QLE growth time from point $x$ to $y$ is symmetric and satisfies the triangle inequality almost surely.
• To establish that the resulting metric is a.s. determined by the quantum surface and the pair of points, and is independent of the choice of approximating sequence.
• To lay the foundational groundwork for proving that the $√{8/3}$-LQG sphere equipped with this metric is equivalent to the Brownian map in law.

Proposed method

• Define $d_{\mathcal{Q}}(x,y)$ as the quantum time it takes for a $\QLE(8/3,0)$ process starting from $x$ to reach $y$ on a $√{8/3}$-LQG surface $\mathcal{S}$.
• Use subsequential limits of discrete approximations to construct the QLE process and show convergence to a well-defined growth process on $\mathcal{S}$.
• Prove that $d_{\mathcal{Q}}(x,y)$ is symmetric a.s. by establishing conditional independence and uniformity of hitting times under reversed exploration.
• Apply a key lemma showing that a non-increasing function $F$ on $[0,D]$ with uniform image must satisfy $F(d) = D - d$, to prove $\tau + \overline{\tau} = D$.
• Use coupling arguments between forward and reverse QLE explorations to show that the sum of growth times equals the total distance, implying the triangle inequality.
• Establish that the metric is a.s. continuous and well-defined on any i.i.d. sequence of points sampled from the quantum area measure.

Experimental results

Research questions

• RQ1 Can a metric structure be canonically defined on the $√{8/3}$-LQG sphere using QLE, such that it satisfies symmetry and the triangle inequality?
• RQ2 Is the QLE-induced distance $d_{\mathcal{Q}}(x,y)$ a.s. symmetric and independent of the choice of approximating sequence?
• RQ3 Does the QLE growth time from $x$ to $y$ agree with the time from $y$ to $x$ in law, and can this be used to prove metric properties?
• RQ4 Can the metric $d_{\mathcal{Q}}$ be extended continuously to the entire $√{8/3}$-LQG surface, and does it match the Brownian map metric?
• RQ5 Is the law of the QLE-induced metric on the $√{8/3}$-LQG sphere consistent with the known properties of the Brownian map?

Key findings

• The QLE-induced distance $d_{\mathcal{Q}}(x,y)$ is a.s. symmetric, satisfying $d_{\mathcal{Q}}(x,y) = d_{\mathcal{Q}}(y,x)$ for almost every pair $(x,y)$.
• The metric $d_{\mathcal{Q}}$ satisfies the triangle inequality almost surely, as shown by coupling forward and reverse QLE explorations and using the identity $\tau + \overline{\tau} = D$.
• The distance $d_{\mathcal{Q}}(x,y)$ is a.s. determined by the triple $(\mathcal{S}, x, y)$, meaning it does not depend on the choice of approximating sequence of points.
• The metric is strictly positive for distinct points, with $d_{\mathcal{Q}}(x,y) > 0$ a.s. for $x \neq y$.
• The construction ensures that $d_{\mathcal{Q}}$ is a.s. a metric on any countable i.i.d. sequence sampled from the quantum area measure.
• The results are consistent with known properties of the Brownian map, supporting the conjecture that the $√{8/3}$-LQG sphere with this metric is equivalent to the Brownian map in law.